Question: Divide the polynomials. The form of your answer should either be $p(x)$ or $p(x)+\dfrac{k}{x+2}$ where $p(x)$ is a polynomial and $k$ is an integer. $\dfrac{3x^3+x^2-4x+12}{x+2}=$
Answer: Usually, there are many different ways to divide polynomials. Here, we will use the method of polynomial long division. $\begin{array}{r} 3x^2-\phantom{1}5x+\phantom{1}6 \\ x+2|\overline{3x^3+\phantom{6}x^2-\phantom{1}4x+12} \\ \mathllap{-(}\underline{3x^3+6x^2\phantom{-14x+12}\rlap )} \\ -5x^2-\phantom{1}4x+12 \\ \mathllap{-(}\underline{-5x^2-10x\phantom{+12}\rlap )} \\ 6x+12 \\ \mathllap{-(}\underline{6x+12\rlap )} \\ 0 \end{array}$ We found that the quotient is $3x^2-5x+6$ and the remainder is $0$, which means the answer is simply a polynomial (no expression of the form $\dfrac{k}{x+2}$ ). $\dfrac{3x^3+x^2-4x+12}{x+2}=3x^2-5x+6$